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The value of $\int \frac{\sqrt{x^2-a^2}}{x}dx$ will be

Let $C$ be a set of $6$ consonants $\{b,c,d,f,g,h\}$ and $V$ be the set of $5$ vowels $\{a,e,i,o,u\}$ and $W$ be the set of seven-letter words that can be formed with these $11$ letters using both the following rules.

$\text{I}.$ The vowels and consonants in the word must alternate.

$\text{II}.$ No letter can be used more than once in a single word.



Then the number of words in the set $W$ is

The coefficient of the term independent of $x$ in the expansion of $(1+x{)}^m{(1+\frac{1}{x})}^n$ is

If function $f\left(x\right)=\{\begin{array}{cc}1,& x=1\\ a^2-3a+x^2,& x>1\end{array}$ has a local maximum at $x=1,$ then the set of values of $a$ is

The sum of $\frac{3}{1\cdot 2}\left(\frac{1}{2}\right)+\frac{4}{2\cdot 3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3\cdot 4}{\left(\frac{1}{2}\right)}^3+\cdots$ upto $20$ terms is

The value of the limit $\underset{\theta \to 0}{lim}\frac{\tan \left(\pi {\cos }^2\theta \right)}{\sin \left(2\pi {\sin }^2\theta \right)}$ is equal to:

If the entry fee to a fair is $\$5$ and each game at the fair costs $\$2$ to play, write an inequality for the total cost when a person plays n games and the maximum amount the person can spend is $\$25$.

$e^{|sinx|}+e^{-|sinx|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$ if

The number of 5 digit numbers thathat can be formed using the digits 0,1,2,3,4,5 that are divisible by 6 when repetition is not allowed?

The content of $3$ urns $1,2,3$ are as follows : $1$ white, $2$ black, $3$ red balls; $2$ white, $1$ black, $1$ red balls; $4$ white, $5$ black, $3$ red balls. One urn is selected at random and two balls are drawn and these comes out to be white and red. Then the probability that they come from

What is the value of $3{\sin }^2{30}^{\circ }+2{\tan }^2{60}^{\circ }-5{\cos }^2{45}^{\circ }$?

A plant is broken by the wind. The branch struck the ground at an angle of ${30}^0$ and at a distance of 30 metres from the root. Find the total height of the plant.

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. then $P^2{R}^3:S^3$ is equal to

The value of $cos\left[2ta{n}^{-1}\right(7\left)\right]$ is

The range of $k$ for which $|\sqrt{x^2+y^2}-\sqrt{x^2+y^2-6x-8y+25}|=k$ will represent hyperbola is

Let $f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$ where $[\cdot ]$ denotes the greatest integer function, then which of the following is true?

If for any square matrix $A^2$ = A, then $(I+A{)}^n$ is equal to (where I is identity matrix)

$\mathrm{If}{D}_r=\left(\begin{array}{ccc}{2}^{r-1}& 2.3^{r-1}& 4.5^{r-1}\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}{\sum }_{r=1}^n{D}_r\mathrm{is}$

A sperical ball of salt is dissolving in water in such a manner that the rate of decrease of its volume at any instant is equal four times its surface area. The rate of decrease in its radius (in units/sec) will be

Let f(x) be a continuous and differentiable function and f(y)f(x + y) = f(x) for all real values of x and y. If f(5) = 3 and f'(3) = 7 then the value of f'(8) is

The coefficient of $x^6.y^{-2}$ in the expansion of ${(\frac{x^2}{y}-\frac{y}{x})}^{12}$ is

Let f: $R\to R$ be defined by$f\left(x\right)=\{\begin{array}{cc}\alpha +\frac{sin\left[x\right]}{2}& ifx>0\\ 2& ifx=0\\ \beta +\left[\frac{sinx-x}{x^3}\right]& ifx<0\end{array}$where $\left[y\right]$ denotes the integral part of y. If f is continuous at $x=0,then\beta -\alpha =$

In four schools $B_1,B_2,B_3,B_4$ the percentage of girls students is 12, 20, 13, 17 respectively. From a school selected at random, one student is picked up at random and it is found that the student is a girl. The probability that the school selected is $B_2$, is

The coefficient of $x^{50}$ in the expansion of $(1+x{)}^{1000}+2x(1+x{)}^{999}+3x^2(1+x{)}^{998}+......+1001x^{1000}$

Find the equation of all lines having
slope 2 which are tangents to the curve.

If $A$ and $B$ are subsets of a set $X,$ then $(A\cap (X-B))\cup B$ is equal to

Question 1

Which of the following is a quadratic equation?

(a) $x^2+2x+1=(4–x{)}^2+3$

(b) $-2x^2=(5-x)(2x-\frac{2}{5})$

(c) $(k+1)x^2+\frac{3}{2}x=7,wherek=-1$

(d) $x^3-x^2=(x-1{)}^3$

A plane which is perpendicular to two planes $2x-2y+z=0$ and $x-y+$ $2z=4$ passes through $(1,-2,1).$ The distance of the plane from the point $(1,2,2)$ is

Let a complex number be $w=1–\sqrt{3}i$. Let another complex number z be such that $|zw|=1$ and $\arg \left(z\right)–\arg \left(w\right)=\frac{\pi }{2}.$ Then the area of the triangle with vertices origin, $z$ and $w$ is equal to: